Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10104 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
246 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,11)(12,19,20,24)(13,18,23,16)(15,22,17,21);(2,5)(3,8)(6,7)(9,11,14,10)(12,17,20,15)(13,16,23,18)(19,22,24,21);(1,2)(3,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,22)(14,20)(16,24)(18,19)(21,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10116, #236438, #236439, … |
| maximal normal subgroup of | C2×#10116, #236438, #236439, … |
| maximal quotient of | #102599, #102600, #102613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3104);